Do you have enough information to work out the area of the shaded quadrilateral?

Mind Your Ps and Qs

Age 16 to 18 Short Challenge Level:

Why do this problem

This
problem will help to train students in the art of careful,
logical, pure thinking which will help to develop their general
mathematical skill. It will require students to address issues
surrounding integration, use of functions, and inequalities,
without needing to go into any particular detail with calculation
of integrals.

Possible approach

Note: This problem might work best
once students have tried Iffy
logic as a starter at a previous time.

Give the students Mind Your
Ps and Qs and let them read it carefully and think about
what it is asking.

Suggest that students discuss in pairs what they think that
the arrow symbols mean. Then, as a group discuss, for example,
why

$$x> 4 \Rightarrow x> 3\mbox{ and } x=-2 \Leftrightarrow
x^3=-8$$

are correct but

$$|x|> 2 \Rightarrow x> 1 \mbox{ and
}x^2=4\Leftrightarrow x=2$$

are incorrect.

The next step is to ensure that everyone can construct their
own individual examples of correct mathematical statements using
propositions from the list. Once students have a couple of examples
of such statements they should share them with the class and
explain their reasoning. Do others agree or disagree? TALKING about
such results will quickly highlight woolly or fallacious thinking
and is an important part of the mathematical process.

Once the group has a feel for constructing the implications,
they need to concentrate on using all of the statements to
construct a complete set of 8 statements. Encourage students to
consider their reasoning clearly in each case. Can the class
complete the task with a clear explanation in each case?

Key questions

What do the arrow symbols mean?

Have you read the question carefully?

Are there certain statementswhich look likely to go
together in a pair?

If an integral is positive or zero, what can we say about the
area enclosed?

Do you remember what the graphs of $\cos$ and $\sin$ look
like?

Possible support

It is rather helpful to draw diagrams and number lines when
thinking about inequalities. Shade the parts of the number line
which apply to a particular inequality to help see which way round
the logic flows.

If possible, start off with Iffy
logic and the support materal suggested there.

Possible extension

Are there multiple solutions? Can students make up a similar set of
questions to give to each other to try? Can they write down really
clear explanations of why, for example, $x> 4\Rightarrow x>
2$?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.